Configurations of lines on del Pezzo surfaces Rosa Winter Universiteit Leiden Konstanz Women in Mathematics Lecture Series June 26th, 2018 I Master ALGANT (see next talk) in Leiden and Padova. I Traineeship 'Eerst de Klas', obtaining a teaching degree and working for a company. I Since 2016: PhD in Leiden under the supervision of Ronald van Luijk and Martin Bright Today: talk about a project that started as my master thesis. A little bit about myself I Bachelor degree from Leiden University I Traineeship 'Eerst de Klas', obtaining a teaching degree and working for a company. I Since 2016: PhD in Leiden under the supervision of Ronald van Luijk and Martin Bright Today: talk about a project that started as my master thesis. A little bit about myself I Bachelor degree from Leiden University I Master ALGANT (see next talk) in Leiden and Padova. I Since 2016: PhD in Leiden under the supervision of Ronald van Luijk and Martin Bright Today: talk about a project that started as my master thesis. A little bit about myself I Bachelor degree from Leiden University I Master ALGANT (see next talk) in Leiden and Padova. I Traineeship 'Eerst de Klas', obtaining a teaching degree and working for a company. Today: talk about a project that started as my master thesis. A little bit about myself I Bachelor degree from Leiden University I Master ALGANT (see next talk) in Leiden and Padova. I Traineeship 'Eerst de Klas', obtaining a teaching degree and working for a company. I Since 2016: PhD in Leiden under the supervision of Ronald van Luijk and Martin Bright A little bit about myself I Bachelor degree from Leiden University I Master ALGANT (see next talk) in Leiden and Padova. I Traineeship 'Eerst de Klas', obtaining a teaching degree and working for a company. I Since 2016: PhD in Leiden under the supervision of Ronald van Luijk and Martin Bright Today: talk about a project that started as my master thesis. Example Cubic surfaces 3 Let's look at smooth cubic surfaces in P over an algebraically closed field. Cubic surfaces 3 Let's look at smooth cubic surfaces in P over an algebraically closed field. Example x3 + y 3 + z3 + 1 = (x + y + z + 1)3 (Clebsch surface) Cubic surfaces 3 Let's look at smooth cubic surfaces in P over an algebraically closed field. Example x3 + y 3 + z3 = 1 (Fermat cubic) Clebsch surface Cubic surfaces Theorem (Cayley-Salmon, 1849) I Such a surface contains exactly 27 lines. I Any point on the surface is contained in at most three of those lines. Cubic surfaces Theorem (Cayley-Salmon, 1849) I Such a surface contains exactly 27 lines. I Any point on the surface is contained in at most three of those lines. Clebsch surface Example The Clebsch surface has 10 Eckardt points; the Fermat cubic has 18 Eckardt points. Cubic surfaces 3 A point on a smooth cubic surface in P that is contained in three lines is called an Eckardt point. Lemma (Hirschfeld, 1967) There are at most 45 Eckardt points on a cubic surface. Cubic surfaces 3 A point on a smooth cubic surface in P that is contained in three lines is called an Eckardt point. Lemma (Hirschfeld, 1967) There are at most 45 Eckardt points on a cubic surface. Example The Clebsch surface has 10 Eckardt points; the Fermat cubic has 18 Eckardt points. Definition A del Pezzo surface X is a 'nice' surface over a field k that has an n embedding in some Pk , such that −aKX is linearly equivalent to a hyperplane section for some a. The degree is the self intersection 2 (−KX ) of the anticanonical divisor. d For degree d ≥ 3, we can embed X as a surface of degree d in P . Question: What do we know about lines on del Pezzo surfaces of other degrees? Generalizations of Eckardt points? More general: del Pezzo surfaces A smooth cubic surface is a surface given by an equation of degree 3 in 3-dimensional space. This is an example of a del Pezzo surface. d For degree d ≥ 3, we can embed X as a surface of degree d in P . Question: What do we know about lines on del Pezzo surfaces of other degrees? Generalizations of Eckardt points? More general: del Pezzo surfaces A smooth cubic surface is a surface given by an equation of degree 3 in 3-dimensional space. This is an example of a del Pezzo surface. Definition A del Pezzo surface X is a 'nice' surface over a field k that has an n embedding in some Pk , such that −aKX is linearly equivalent to a hyperplane section for some a. The degree is the self intersection 2 (−KX ) of the anticanonical divisor. Question: What do we know about lines on del Pezzo surfaces of other degrees? Generalizations of Eckardt points? More general: del Pezzo surfaces A smooth cubic surface is a surface given by an equation of degree 3 in 3-dimensional space. This is an example of a del Pezzo surface. Definition A del Pezzo surface X is a 'nice' surface over a field k that has an n embedding in some Pk , such that −aKX is linearly equivalent to a hyperplane section for some a. The degree is the self intersection 2 (−KX ) of the anticanonical divisor. d For degree d ≥ 3, we can embed X as a surface of degree d in P . More general: del Pezzo surfaces A smooth cubic surface is a surface given by an equation of degree 3 in 3-dimensional space. This is an example of a del Pezzo surface. Definition A del Pezzo surface X is a 'nice' surface over a field k that has an n embedding in some Pk , such that −aKX is linearly equivalent to a hyperplane section for some a. The degree is the self intersection 2 (−KX ) of the anticanonical divisor. d For degree d ≥ 3, we can embed X as a surface of degree d in P . Question: What do we know about lines on del Pezzo surfaces of other degrees? Generalizations of Eckardt points? We often do this to resolve a singularity. Another way of defining del Pezzo surfaces Let P be a point in the plane. The construction blowing up replaces P by a line E, called the exceptional curve above P; each point on this line E is identified with a direction through P. Another way of defining del Pezzo surfaces Let P be a point in the plane. The construction blowing up replaces P by a line E, called the exceptional curve above P; each point on this line E is identified with a direction through P. We often do this to resolve a singularity. From: Robin Hartshorne, Algebraic Geometry. I We say that X lies above the plane. I On X (so after blowing up), P is no longer a point, but a line. I Two lines that intersect in the plane in P do not intersect in X ! They both intersect the exceptional curve E, but in different points. I Outside P, everything stays the same. Some facts about blow-ups of points Let P be a point in the plane that we blow up, and let E be the exceptional curve above P. We call the resulting surface X . I On X (so after blowing up), P is no longer a point, but a line. I Two lines that intersect in the plane in P do not intersect in X ! They both intersect the exceptional curve E, but in different points. I Outside P, everything stays the same. Some facts about blow-ups of points Let P be a point in the plane that we blow up, and let E be the exceptional curve above P. We call the resulting surface X . I We say that X lies above the plane. I Two lines that intersect in the plane in P do not intersect in X ! They both intersect the exceptional curve E, but in different points. I Outside P, everything stays the same. Some facts about blow-ups of points Let P be a point in the plane that we blow up, and let E be the exceptional curve above P. We call the resulting surface X . I We say that X lies above the plane. I On X (so after blowing up), P is no longer a point, but a line. I Outside P, everything stays the same. Some facts about blow-ups of points Let P be a point in the plane that we blow up, and let E be the exceptional curve above P. We call the resulting surface X . I We say that X lies above the plane. I On X (so after blowing up), P is no longer a point, but a line. I Two lines that intersect in the plane in P do not intersect in X ! They both intersect the exceptional curve E, but in different points. Some facts about blow-ups of points Let P be a point in the plane that we blow up, and let E be the exceptional curve above P. We call the resulting surface X . I We say that X lies above the plane. I On X (so after blowing up), P is no longer a point, but a line. I Two lines that intersect in the plane in P do not intersect in X ! They both intersect the exceptional curve E, but in different points. I Outside P, everything stays the same. Theorem Let X be a del Pezzo surface of degree d over an algebraically closed field.